Mathematical Sciences: Stochastic Models for Reliability of Systems with Dependencies Among Components

提案：DMS 9503104研究员：W. J. Padgett， J. D. Lynch和S. D. Durham研究所：南卡罗莱纳大学哥伦比亚分校题目：具有组件间依赖关系的系统可靠性的随机模型摘要：研究涉及复杂系统的可靠性。有三个主要的重点：(1)组件可靠性建模，(2)将组件可靠性和依赖关系纳入系统可靠性，从而产生易于处理的数据分析模型，以及(3)为这些模型开发临界理论。对于(1)，研究了一些用于构件可靠性分析的模型，包括条件威布尔分布和反高斯分布以及泊松-威布尔缺陷模型。研究了“条件威布尔”分布，因为它很好地拟合某些纤维强度数据集，并且由于纤维制造和测试过程中的“审查”考虑可能是合理的。有限状态马尔可夫随机强度泊松-威布尔缺陷模型以混合分布（零强度）和混合风险模型（无限强度）为极值。这部分项目的一个主要目标是研究该缺陷模型的统一混合理论，该理论分别包含为混合分布的极值和混合危害开发的混合理论。为了解决(2)和(3)，研究了一个一般模型，该模型本质上是分层的。层次结构包括(i)微观（或组件）级，（ii）子系统（或“组件束”）级和（iii）系统（或“束链”）级。人们早就知道，在许多系统中，一个部件的失效会改变施加在其余部件上的应力。因此，以一种现实的方式将组件依赖关系合并到可靠性模型中对于获得准确的结果是非常可取的。在这里，通过使用“负载共享规则”将组件依赖/交互合并到模型中，这最适用于由于机械或物理考虑而导致“加载”的情况。考虑了一般单调负荷分担规则，提出了系统可靠度的计算方法。目前的研究包括一类特殊的规则，其中可以使用网络上随机行走的吸收概率来计算组件负载。特别是，一些流行的负荷分担规则可以简化为电网的考虑。当构件失效时，对能量的考虑使我们深入了解由这些规则引起的应力集中的行为。主要目标是继续研究这些电气网络规则，以及适用于机械负载传递情况的其他网络规则，特别是复合材料，特别是研究负载可以通过基体材料传递的“有效距离”及其与纤维断裂周围的“无效长度”的关系，以便通过基体传递负载。还考虑了临界性问题。对于k- of-n系统和具有少量组件的系统的初步工作表明，当每个组件使用混合变换伽玛模型具有威布尔失效分布时，可以获得精确的理论，其中转换取决于威布尔分布。该模型还适用于极值近似误差的计算和混合分布的系统辨识。该研究涉及描述具有依赖组件的复杂系统失效的模型的开发。研究人员正在研究复杂系统的可靠性模型，包括允许组件依赖和相互作用的复杂材料。这些模型可以直接应用于许多当前重要的问题，包括纤维复合材料和电网的失效。这对高可靠性复杂材料的设计和大规模制造具有重要意义。

Proposal: DMS 9503104 PIs: W. J. Padgett, J. D. Lynch and S. D. Durham Institution: University of South Carolina - Columbia Title: STOCHASTIC MODELS FOR RELIABILITY OF SYSTEMS WITH DEPENDENCIES AMONG COMPONENTS Abstract: The research involves the reliability of complex systems. There are three main thrusts: (1) modeling component reliability, (2) incorporating component reliability and dependencies into the system reliability which result in tractable data analytic models, and (3) developing a criticality theory for such models. Regarding (1), some models are studied for the analysis of component reliability, including the conditional Weibull and inverse Gaussian distributions and a Poisson-Weibull flaw model. The "conditional Weibull" distribution is studied since it fits certain fiber strength data sets well and may be justified due to "censoring" considerations in the fiber manufacturing and testing processes. The Poisson-Weibull flaw model with finite-state Markov random intensity has the mixed distribution (zero intensity) and the mixed hazard model (infinite intensity) as extremes. A major objective of this part of the project is the investigation of a unified mixture theory for this flaw model which subsumes the theory of mixtures developed for the extremes of mixed distributions and mixed hazards, respectively. To address (2) and (3), a general model is investigated which is hierarchical in nature. The hierarchy consists of (i) the micro (or component) level, (ii) the subsystem (or "bundle of components") level, and (iii) the system (or "chain of bundles") level. It has long been known that in many systems, the failure of a component changes the stress applied to the remaining components. Thus, incorporating component dependencies into a reliability model in a realistic manner is highly desirable for accurate results. Here, component dependencies/interactions are incorporated into the model by using "load-sharing rules," m ost applicable to situations where "loadings" are due to mechanical or physical considerations. General monotone load-sharing rules are considered for which a method of calculating the system reliability has been developed. The present research includes a special class of these rules where the component load can be calculated using absorption probabilities for random walks on a network. In particular, a number of the popular load-sharing rules can be reduced to the consideration of electrical networks. Energy considerations give insight into the behavior of stress concentrations induced by these rules as components fail. Major objectives are to continue the investigation of these electrical network rules, and other network rules which are appropriate for mechanical load transfer situations, especially for composite materials, and, specifically to study the "effective distance" that a load can be transferred via matrix material and its relationship to the "ineffective length" around fiber breaks for load transfer through the matrix. Criticality issues are also considered. Preliminary work for k-out-of-n systems and systems with a small number of components suggest that an exact theory may be obtainable when each component has a Weibull failure distribution using a mixed transformed gamma model, where the transformation depends on the Weibull distribution. Such a model also lends itself to calculation of extreme value approximation errors and to system identification using the mixing distribution. The research involves the development of models for describing the failure of complex systems with dependent components. The researchers are investigating reliability models for complex systems of components including complex materials which allow for component dependencies and interactions. Such models have direct application to many problems of current importance, including failure of fibrous composite materials and electrical networks. This has important imp lications for the design and large scale manufacture of complex materials where high reliability.